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It is well-known that for a harmonic function $u$ defined on the unit ball ofthe $d$-dimensional Euclidean space, $d\geq 2$, the tangential and normalcomponent of the gradient $\nabla u$ on the sphere are comparable by means ofthe $L^p$-norms, $p\in(1,\infty)$, up to multiplicative constants that dependonly on $d,p$. This paper formulates and proves a discrete analogue of thisresult for discrete harmonic functions defined on a discrete box on the$d$-dimensional lattice with multiplicative constants that do not depend on thesize of the box.

An $L^p$-comparison, $p\in (1,\infty)$, on the finite differences of a discrete harmonic function at the boundary of a discrete box